Euler–Lagrange–Herglotz equations on Lie algebroids

We introduce Euler–Lagrange–Herglotz equations on Lie algebroids. The methodology is to extend the Jacobi structure from TQ×ℝ and T^*Q ×ℝ to A×ℝ and A^*×ℝ , respectively, where A is a Lie algebroid and A^* carries the associated Poisson structure. We see that A^*×ℝ possesses a natural Jacobi structure from where we are able to model dissipative mechanical systems on Lie algebroids, generalizing previous models on TQ×ℝ and introducing new ones as for instance for reduced systems on Lie algebras, semidirect products (action Lie algebroids) and Atiyah bundles.